Communications of the ACM
A Comment on the Efficiency of Secret Sharing Scheme over Any Finite Abelian Group
ACISP '98 Proceedings of the Third Australasian Conference on Information Security and Privacy
Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
CRYPTO '89 Proceedings of the 9th Annual International Cryptology Conference on Advances in Cryptology
Optimal-resilience proactive public-key cryptosystems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Some results in linear secret sharing
Some results in linear secret sharing
Practical threshold signatures
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Efficient multi-party computation over rings
EUROCRYPT'03 Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques
Non-interactive Proofs for Integer Multiplication
EUROCRYPT '07 Proceedings of the 26th annual international conference on Advances in Cryptology
Revisiting the Karnin, Greene and Hellman Bounds
ICITS '08 Proceedings of the 3rd international conference on Information Theoretic Security
Algebraic geometric secret sharing schemes and secure multi-party computations over small fields
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Linear integer secret sharing and distributed exponentiation
PKC'06 Proceedings of the 9th international conference on Theory and Practice of Public-Key Cryptography
Efficient integer span program for hierarchical threshold access structure
Information Processing Letters
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A black-box secret sharing scheme (BBSSS) for a given access structure works in exactly the same way over any finite Abelian group, as it only requires black-box access to group operations and to random group elements. In particular, there is no dependence on e.g. the structure of the group or its order. The expansion factor of a BBSSS is the length of a vector of shares (the number of group elements in it) divided by the number of players n. At CRYPTO 2002 Cramer and Fehr proposed a threshold BBSSS with an asymptotically minimal expansion factor Θ(log n). In this paper we propose a BBSSS that is based on a new paradigm, namely, primitive sets in algebraic number fields. This leads to a new BBSSS with an expansion factor that is absolutely minimal up to an additive term of at most 2, which is an improvement by a constant additive factor. We provide good evidence that our scheme is considerably more efficient in terms of the computational resources it requires. Indeed, the number of group operations to be performed is Õ(n2) instead of Õ(n3) for sharing and Õ(n1.6) instead of Õ(n2.6) for reconstruction. Finally, we show that our scheme, as well as that of Cramer and Fehr, has asymptotically optimal randomness efficiency.